What time of year are you thinking of?
Regarding accommodation: As I mentioned above, there are options outside the park (Field) or in the park in B&Bs. I was able to find reasonably priced accommodation for early/mid-June all right.
Regarding hostels: Book early. I tried to get hostel accommodation, but it was all booked up by the time I started looking ... and I was looking for shoulder season accommodation, not peak season.
Regarding the climbs: They're not that bad. Long climbs like the ones in the Rockies are easier than shorter, steeper climbs.
Regarding the weather: Chances are you'd have great weather in July ... most likely. In any other month, it'll be anyone's guess. This year, it snowed quite heavily on the May long weekend (end of May) ... and quite often it snows in August. But sometimes it doesn't. However, be prepared for rain, snow, sleet, hail, and quite hot temps.
Regarding a good month to travel: While July will likely have the best weather, it is also the busiest month. It can be quite a zoo! Early to mid-June is quieter ... as is late August. But the weather isn't as stable then.
Regarding supported tours: You can go that route (I never have) but it is not that difficult to tour that area. I've cycled from Jasper to Golden in one day ... now, granted, I was trained for long distance cycling and that was in the middle of the Rocky Mountain 1200, but still, I wasn't the greatest climber in the world or the fastest cyclist in the world. I've also cycled it quite comfortably in 3 days, and I've cycled fairly lengthy portions of it on day rides many times.